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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(967, base_ring=CyclotomicField(966))
M = H._module
chi = DirichletCharacter(H, M([724]))
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pari:[g,chi] = znchar(Mod(59,967))
| Modulus: | \(967\) | |
| Conductor: | \(967\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(483\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
|
| Real: | no |
| Primitive: | yes |
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sage:chi.is_primitive()
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pari:#znconreyconductor(g,chi)==1
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{967}(2,\cdot)\)
\(\chi_{967}(4,\cdot)\)
\(\chi_{967}(16,\cdot)\)
\(\chi_{967}(18,\cdot)\)
\(\chi_{967}(21,\cdot)\)
\(\chi_{967}(22,\cdot)\)
\(\chi_{967}(25,\cdot)\)
\(\chi_{967}(31,\cdot)\)
\(\chi_{967}(32,\cdot)\)
\(\chi_{967}(34,\cdot)\)
\(\chi_{967}(35,\cdot)\)
\(\chi_{967}(36,\cdot)\)
\(\chi_{967}(44,\cdot)\)
\(\chi_{967}(49,\cdot)\)
\(\chi_{967}(50,\cdot)\)
\(\chi_{967}(57,\cdot)\)
\(\chi_{967}(59,\cdot)\)
\(\chi_{967}(60,\cdot)\)
\(\chi_{967}(65,\cdot)\)
\(\chi_{967}(70,\cdot)\)
\(\chi_{967}(83,\cdot)\)
\(\chi_{967}(84,\cdot)\)
\(\chi_{967}(91,\cdot)\)
\(\chi_{967}(98,\cdot)\)
\(\chi_{967}(101,\cdot)\)
\(\chi_{967}(103,\cdot)\)
\(\chi_{967}(106,\cdot)\)
\(\chi_{967}(111,\cdot)\)
\(\chi_{967}(114,\cdot)\)
\(\chi_{967}(115,\cdot)\)
...
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sage:chi.galois_orbit()
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pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\(5\) → \(e\left(\frac{362}{483}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 967 }(59, a) \) |
\(1\) | \(1\) | \(e\left(\frac{148}{483}\right)\) | \(e\left(\frac{101}{161}\right)\) | \(e\left(\frac{296}{483}\right)\) | \(e\left(\frac{362}{483}\right)\) | \(e\left(\frac{451}{483}\right)\) | \(e\left(\frac{104}{483}\right)\) | \(e\left(\frac{148}{161}\right)\) | \(e\left(\frac{41}{161}\right)\) | \(e\left(\frac{9}{161}\right)\) | \(e\left(\frac{87}{161}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
